We can represent more complicated expressions such as \[\begin{bmatrix} w\\x\\y\\z \end{bmatrix} =\begin{bmatrix} \sqrt{a-\sqrt{b-\sqrt{c-\sqrt{d}}}}\\ \sqrt{b-\sqrt{c-\sqrt{d}}}\\ \sqrt{c-\sqrt{d}}\\ \sqrt{d} \end{bmatrix}\] we may write a chain of transformations (compositions) \[\begin{bmatrix} w\\x\\y\\z \end{bmatrix} = \begin{bmatrix} \sqrt{\cdot} & 0 & 0 & 0 \\ 0 & \sqrt{\cdot} & 0 & 0 \\ 0 & 0 & \sqrt{\cdot} & 0 \\ 0 & 0 & 0 & \sqrt{\cdot} \\ \end{bmatrix} \odot \left( \begin{bmatrix} 1 & -\sqrt{\cdot} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \odot \left( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & -\sqrt{\cdot} & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \odot \left( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -\sqrt{\cdot} \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \odot \begin{bmatrix} a\\b\\c\\d \end{bmatrix} \right)\right)\right)\]
this example is the solution to the set of equations \[\begin{bmatrix} \cdot^2 & 1 & 0 & 0 \\ 0 & \cdot^2 & 1 & 0 \\ 0 & 0 & \cdot^2 & 1 \\ 0 & 0 & 0 & \cdot^2 \\ \end{bmatrix} \odot \begin{bmatrix} w\\x\\y\\z \end{bmatrix} = \begin{bmatrix} a\\b\\c\\d \end{bmatrix}\] noting that one transformation matrix is required per additional \(1\) in the initial statement of the equations.